Systolic Geometry and Topology

Share this page

Mikhail G. Katz

The systole of a compact metric space
\(X\) is a metric invariant of \(X\), defined as the
least length of a noncontractible loop in \(X\). When
\(X\) is a graph, the invariant is usually referred to as the
girth, ever since the 1947 article by W. Tutte. The first nontrivial
results for systoles of surfaces are the two classical inequalities of
C. Loewner and P. Pu, relying on integral-geometric identities, in the
case of the two-dimensional torus and real projective plane,
respectively. Currently, systolic geometry is a rapidly developing
field, which studies systolic invariants in their relation to other
geometric invariants of a manifold.

This book presents the systolic geometry of manifolds and polyhedra,
starting with the two classical inequalities, and then proceeding to recent
results, including a proof of M. Gromov's filling area conjecture in a
hyperelliptic setting. It then presents Gromov's inequalities and their
generalisations, as well as asymptotic phenomena for systoles of surfaces of
large genus, revealing a link both to ergodic theory and to properties of
congruence subgroups of arithmetic groups. The author includes results on
the systolic manifestations of Massey products, as well as of the classical
Lusternik-Schnirelmann category.

Readership

Graduate students and research mathematicians interested in
new methods in differential geometry and topology.

The systole of a compact metric space
\(X\) is a metric invariant of \(X\), defined as the
least length of a noncontractible loop in \(X\). When
\(X\) is a graph, the invariant is usually referred to as the
girth, ever since the 1947 article by W. Tutte. The first nontrivial
results for systoles of surfaces are the two classical inequalities of
C. Loewner and P. Pu, relying on integral-geometric identities, in the
case of the two-dimensional torus and real projective plane,
respectively. Currently, systolic geometry is a rapidly developing
field, which studies systolic invariants in their relation to other
geometric invariants of a manifold.

This book presents the systolic geometry of manifolds and polyhedra,
starting with the two classical inequalities, and then proceeding to recent
results, including a proof of M. Gromov's filling area conjecture in a
hyperelliptic setting. It then presents Gromov's inequalities and their
generalisations, as well as asymptotic phenomena for systoles of surfaces of
large genus, revealing a link both to ergodic theory and to properties of
congruence subgroups of arithmetic groups. The author includes results on
the systolic manifestations of Massey products, as well as of the classical
Lusternik-Schnirelmann category.